# Difference between revisions of "Locally finite semi-group"

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− | A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a [[Periodic semi-group|periodic semi-group]] (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see [[Burnside problem|Burnside problem]]). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. [[Nil semi-group|Nil semi-group]]). These are, for example, a free semi-group with two generators in the variety given by | + | {{TEX|done}} |

+ | A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a [[Periodic semi-group|periodic semi-group]] (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see [[Burnside problem|Burnside problem]]). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. [[Nil semi-group|Nil semi-group]]). These are, for example, a free semi-group with two generators in the variety given by $x^3=0$, and a free semi-group with three generators in the variety given by $x^2=0$. Moreover, for a number of classes of semi-groups the conditions of periodicity and local finiteness are equivalent. A trivial example is given by commutative semi-groups. A band of locally finite semi-groups (see [[Band of semi-groups|Band of semi-groups]]) is itself a locally finite semi-group [[#References|[1]]]; moreover, a semi-group that has a decomposition into locally finite groups is a locally finite semi-group; in particular, a semi-group of idempotents (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]) is a locally finite semi-group [[#References|[7]]]. If $n$ is such that any group satisfying the law $x^n=1$ is locally finite, then any semi-group with the law $x^{n+1}=x$ is locally finite [[#References|[6]]]. A semi-group that has a decomposition into locally finite semi-groups need not be a locally finite semi-group [[#References|[3]]], but if $\rho$ is a congruence on a semi-group $S$ such that the quotient semi-group $S/\rho$ is locally finite and every $\rho$-class that is a sub-semi-group is locally finite, then $S$ is a locally finite semi-group (see [[#References|[4]]], [[#References|[5]]]); in particular, an ideal extension of a locally finite semi-group by a locally finite semi-group is itself a locally finite semi-group. If $S$ is a periodic semi-group of matrices over a skew-field and all subgroups of $S$ are locally finite, then $S$ is locally finite [[#References|[8]]], which implies that any periodic semi-group of matrices over an arbitrary field is locally finite. | ||

− | If | + | If $S$ is a periodic inverse semi-group of matrices over a field and, moreover, the periods of all its elements (see [[Monogenic semi-group|Monogenic semi-group]]) are uniformly bounded and are not divided by the characteristic of the field, then $S$ is finite [[#References|[2]]]. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.N. Shevrin, "On locally finite semigroups" ''Soviet Math. Dokl.'' , '''6''' (1965) pp. 769–772 ''Dokl. Akad. Nauk SSSR'' , '''162''' (1965) pp. 770–773</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.B. Shneperman, "Periodic inverse linear semigroups" ''Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk.'' , '''4''' (1976) pp. 22–28 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T.C. Brown, "On locally finite semigroups" ''Ukr. Math. J.'' , '''20''' (1968) pp. 631–636 ''Ukr. Mat. Zh.'' , '''20''' : 6 (1968) pp. 732–738</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T.C. Brown, "A semigroup union of disjoint locally finite subsemigroups which is not locally finite" ''Pacific J. Math.'' , '''22''' : 1 (1967) pp. 11–14</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> T.C. Brown, "An interesting combinatorial method in the theory of locally finite semigroups" ''Pacific J. Math.'' , '''36''' : 2 (1971) pp. 285–289</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.A. Green, D. Rees, "On semi-groups in which | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.N. Shevrin, "On locally finite semigroups" ''Soviet Math. Dokl.'' , '''6''' (1965) pp. 769–772 ''Dokl. Akad. Nauk SSSR'' , '''162''' (1965) pp. 770–773</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.B. Shneperman, "Periodic inverse linear semigroups" ''Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk.'' , '''4''' (1976) pp. 22–28 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> T.C. Brown, "On locally finite semigroups" ''Ukr. Math. J.'' , '''20''' (1968) pp. 631–636 ''Ukr. Mat. Zh.'' , '''20''' : 6 (1968) pp. 732–738</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T.C. Brown, "A semigroup union of disjoint locally finite subsemigroups which is not locally finite" ''Pacific J. Math.'' , '''22''' : 1 (1967) pp. 11–14</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> T.C. Brown, "An interesting combinatorial method in the theory of locally finite semigroups" ''Pacific J. Math.'' , '''36''' : 2 (1971) pp. 285–289</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.A. Green, D. Rees, "On semi-groups in which $x^r=x$" ''Proc. Cambridge Philos. Soc.'' , '''48''' : 1 (1952) pp. 35–40</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. McLean, "Idempotent semigroups" ''Amer. Math. Monthly'' , '''61''' : 2 (1954) pp. 110–113</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R. McNaughton, Y. Zalcstein, "The Burnside problem for semigroups" ''J. of Algebra'' , '''34''' : 2 (1975) pp. 292–299</TD></TR></table> |

## Latest revision as of 08:27, 3 October 2014

A semi-group in which every finitely-generated sub-semi-group is finite. A locally finite semi-group is a periodic semi-group (torsion semi-group). The converse is false: There are even torsion groups that are not locally finite (see Burnside problem). Long before the solution of the Burnside problem for groups, examples had been constructed of semi-groups that are torsion but not locally finite in classes of semi-groups remote from groups; above all, in the class of nil semi-groups (cf. Nil semi-group). These are, for example, a free semi-group with two generators in the variety given by $x^3=0$, and a free semi-group with three generators in the variety given by $x^2=0$. Moreover, for a number of classes of semi-groups the conditions of periodicity and local finiteness are equivalent. A trivial example is given by commutative semi-groups. A band of locally finite semi-groups (see Band of semi-groups) is itself a locally finite semi-group [1]; moreover, a semi-group that has a decomposition into locally finite groups is a locally finite semi-group; in particular, a semi-group of idempotents (cf. Idempotents, semi-group of) is a locally finite semi-group [7]. If $n$ is such that any group satisfying the law $x^n=1$ is locally finite, then any semi-group with the law $x^{n+1}=x$ is locally finite [6]. A semi-group that has a decomposition into locally finite semi-groups need not be a locally finite semi-group [3], but if $\rho$ is a congruence on a semi-group $S$ such that the quotient semi-group $S/\rho$ is locally finite and every $\rho$-class that is a sub-semi-group is locally finite, then $S$ is a locally finite semi-group (see [4], [5]); in particular, an ideal extension of a locally finite semi-group by a locally finite semi-group is itself a locally finite semi-group. If $S$ is a periodic semi-group of matrices over a skew-field and all subgroups of $S$ are locally finite, then $S$ is locally finite [8], which implies that any periodic semi-group of matrices over an arbitrary field is locally finite.

If $S$ is a periodic inverse semi-group of matrices over a field and, moreover, the periods of all its elements (see Monogenic semi-group) are uniformly bounded and are not divided by the characteristic of the field, then $S$ is finite [2].

#### References

[1] | L.N. Shevrin, "On locally finite semigroups" Soviet Math. Dokl. , 6 (1965) pp. 769–772 Dokl. Akad. Nauk SSSR , 162 (1965) pp. 770–773 |

[2] | L.B. Shneperman, "Periodic inverse linear semigroups" Vesci Akad. Nauk BSSR Ser. Fiz. Mat. Nauk. , 4 (1976) pp. 22–28 (In Russian) |

[3] | T.C. Brown, "On locally finite semigroups" Ukr. Math. J. , 20 (1968) pp. 631–636 Ukr. Mat. Zh. , 20 : 6 (1968) pp. 732–738 |

[4] | T.C. Brown, "A semigroup union of disjoint locally finite subsemigroups which is not locally finite" Pacific J. Math. , 22 : 1 (1967) pp. 11–14 |

[5] | T.C. Brown, "An interesting combinatorial method in the theory of locally finite semigroups" Pacific J. Math. , 36 : 2 (1971) pp. 285–289 |

[6] | J.A. Green, D. Rees, "On semi-groups in which $x^r=x$" Proc. Cambridge Philos. Soc. , 48 : 1 (1952) pp. 35–40 |

[7] | D. McLean, "Idempotent semigroups" Amer. Math. Monthly , 61 : 2 (1954) pp. 110–113 |

[8] | R. McNaughton, Y. Zalcstein, "The Burnside problem for semigroups" J. of Algebra , 34 : 2 (1975) pp. 292–299 |

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Locally finite semi-group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_finite_semi-group&oldid=33470